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Azimuthal and Particle in a box equations

  • Thread starter maxiee
  • Start date
  • #1
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Homework Statement


Mathematically, the Azimuthal equation is the same differential equation as the one for a particle in a box. But [tex] \Phi(\phi) [/tex] for [tex] m_l = 0 [/tex], is a constant and is allowed, whereas such a constant wave function is not allowed for a particle in a box. What physics accounts for the difference?


Homework Equations


The Azimuthal Equation:
[tex]
\frac{\partial ^{2} \Phi(\phi)}{\partial \phi^{2}} = -m_l ^{2} \Phi(\phi)
[/tex]

The particle in a box equation:
[tex]
\frac{\partial ^{2} \psi(x)}{\partial \psi^{2}} = -k ^{2} \psi(x)
[/tex]


The Attempt at a Solution


The boundary conditions seem to play a role in the different allowed wave functions. However, I am having trouble relating the boundary conditions to the allowed quantum numbers.

Thanks in advance

Edit: The Azimuthal Equation corresponds to the Azimuthal motion of a particle. It comes about from the 3D Schrodinger Eq.
The Equation for a particle in a box is the result of the 1D Schrodinger Eq.
 
Last edited:

Answers and Replies

  • #2
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There may come restrictions on [tex]m_l[/tex] or [tex]k[/tex] from the differential equations for the other coordinates. Now, I don't really remember the details but I think the radial equation imposes constraints on [tex]m_l[/tex]. The notation implies that [tex]m_l=m_l(l)[/tex]. That is, [tex]m_l[/tex] is a function of [tex]l[/tex], which appears in the equation for [tex]\theta[/tex] I think, solved by the Legendre polynomials. I might be remembering this wrong however, mixing things up. But in general, I think this would be the difference - that you have different constraints in the two cases.
[STRIKE]
It would probably be good if you could explain a bit more in detail where the equations come from, I can only assume you are solving some PDE similar to the wave eq. or Schrödinger eq.[/STRIKE]

EDIT: Sorry, I can see that it is most probably the Schrödinger eq. you are solving.
 
Last edited:

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